For an ASTM $30$ cast iron, $($a$)$ find the factors of safety using the BCM and MM theories, $($b$)$ plot the failure diagrams in the sigma$\_$a$,$ sigma$\_$b plane to scale and locate the coordinates of the stress state, and $($c$)$ estimate the factors of safety from the two theories by graphical measurements along the load line.

$\backslash$table$[[$Problem Number,$\backslash$sigma $\_($x$)($k$\backslash$psi $),\backslash$sigma $\_($y$)($k$\backslash$psi $),\backslash$tau $\_($xy$)($k$\backslash$psi $)],[5-21,15,10,0],[5-22,15,-50,0],[5-23,15,0,-10],[5-24,-10,-25,-10],[5-25,-35,13,-10]]$